|
%I M5390 #127 Feb 20 2024 08:24:39
%S 1,127,149,251,331,337,373,509,599,701,757,809,877,905,907,959,977,
%T 997,1019,1087,1199,1207,1211,1243,1259,1271,1477,1529,1541,1549,1589,
%U 1597,1619,1649,1657,1719,1759,1777,1783,1807,1829,1859,1867,1927,1969,1973,1985,2171,2203,2213,2231,2263,2279,2293,2377,2429,2465,2503,2579,2669
%N Odd numbers not of form p + 2^k (de Polignac numbers).
%C Contains both primes (A065381) and composites (A098237). - _Jonathan Vos Post_, Jun 19 2008
%C Crocker shows that this sequence is infinite; in particular, 2^2^n - 5 is in this sequence for each n > 2. - _Charles R Greathouse IV_, Sep 01 2015
%C Problem: what is the asymptotic density of de Polignac numbers? Based on the data in A254248, it seems this sequence may have an asymptotic density d > 0.05. Conjecture (cf. Pomerance 2013): the density d(n) of de Polignac numbers <= n is d(n) ~ (1 - 2/log(n))^(log(n)/log(2)), so the asymptotic density d = exp(-2/log(2)) = 0.055833... = 0.111666.../2. - _Thomas Ordowski_, Jan 30 2021
%C From _Amiram Eldar_, Feb 03 2021: (Start)
%C Romanov (or Romanoff) proved in 1934 that the complementary sequence has a positive lower asymptotic density, and the assumed asymptotic density was later named Romanov's constant (Pintz, 2006).
%C The lower asymptotic density of this sequence is positive (Van Der Corput, 1950; Erdős, 1950), and larger than 0.00905 (Habsieger and Roblot, 2006).
%C The upper asymptotic density of this sequence is smaller than 0.392352 (Elsholtz and Schlage-Puchta, 2018).
%C Previous bounds on the upper asymptotic density were given by Chen and Sun (2006), Pintz (2006), Habsieger and Roblot (2006), Lü (2007) and Habsieger and Sivak-Fischler (2010).
%C Romani (1983) conjectured that the asymptotic density of this sequence is 0.066... (End)
%D Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Inc., NJ, 2005, pp. 62 & 300.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D J. G. Van Der Corput, On de Polignac's conjecture, Simon Stevin, Vol. 27 (1950), pp. 99-105.
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, see #127.
%H T. D. Noe, <a href="/A006285/b006285.txt">Table of n, a(n) for n = 1..10000</a>
%H Yong-Gao Chen and Xue-Gong Sun, <a href="https://doi.org/10.1016/j.jnt.2003.11.009">On Romanoff's constant</a>, Journal of Number Theory, Vol. 106, No. 2 (2004), pp. 275-284.
%H Roger Crocker, <a href="http://www.jstor.org/stable/2688349">A theorem concerning prime numbers</a>, Mathematics Magazine, Vol. 34, No. 6 (1961), pp. 316-344.
%H Yuchen Ding, <a href="https://arxiv.org/abs/2201.12783">On a problem of Romanoff type</a>, arXiv:2201.12783 [math.NT], 2022.
%H Christian Elsholtz and Jan-Christoph Schlage-Puchta, <a href="https://doi.org/10.1007/s00209-017-1908-x">On Romanov's constant</a>, Mathematische Zeitschrift, Vol. 288 (2018), pp. 713-724; <a href="https://www.researchgate.net/profile/Jan_Christoph_Schlage-Puchta/publication/317759102_On_Romanov%27s_constant">alternative link</a>.
%H Paul Erdős, <a href="https://users.renyi.hu/~p_erdos/1950-07.pdf">On integers of the form 2^k + p and some related problems</a>, Summa Brasil. Math., Vol. 2 (1950), p. 113-125.
%H Laurent Habsieger and Xavier-François Roblot, <a href="https://hal.archives-ouvertes.fr/hal-00863145/">On integers of the form p+2^k</a>, Acta Arithmetica, Vol. 122, No. 1 (2006), pp. 45-50.
%H Laurent Habsieger and Jimena Sivak-Fischler, <a href="https://doi.org/10.1007/s00013-010-0202-5">An effective version of the Bombieri-Vinogradov theorem, and applications to Chen's theorem and to sums of primes and powers of two</a>, Archiv der Mathematik, Vol. 95, No. 6 (2010), pp. 557-566.
%H Guang-Shi Lü, <a href="http://caod.oriprobe.com/articles/11909183/On_Romanoff_s_Constant_and_Its_Generalized_Problem.htm">On Romanoff's constant and its generalized problem</a>, Chinese Advances in Mathematics, Vol. 36, No. 1 (2007), pp. 94-100.
%H János Pintz, <a href="http://doi.org/10.1007/s10474-006-0060-6">A note on Romanov's constant</a>, Acta Mathematica Hungarica, Vol. 112, No. 1-2 (2006), pp. 1-14.
%H Paul Pollack, <a href="http://pollack.uga.edu/NABDofficial.pdf">Not Always Buried Deep: A Second Course in Elementary Number Theory</a>, AMS, 2009, p. 201, exercise 34.
%H Carl Pomerance, <a href="https://math.dartmouth.edu/~carlp/coverbirthtalk.pdf">Erdős, van der Corput, and the birth of covering congruences</a>, Joint Mathematics Meetings, Special Session on Covering Congruences, San Diego, CA, January, 2013.
%H F. Romani, <a href="https://doi.org/10.1007/BF02576468">Computations concerning primes and powers of two</a>, Calcolo, Vol. 20 (1983), pp. 319-336.
%H Nikolai Pavlovich Romanoff, <a href="https://doi.org/10.1007/BF01449161">Über einige Sätze der additiven Zahlentheorie</a>, Math. Ann., Vol. 109 (1934), pp. 668-678.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Romanov%27s_theorem">Romanov's theorem</a>.
%F A109925(a(n)) = 0. - _Reinhard Zumkeller_, May 27 2015
%F Conjecture: a(n) ~ n*exp(2/log(2)) = n*17.91... - _Thomas Ordowski_, Feb 02 2021
%e 127 is in the sequence since 127 - 2^0 = 126, 127 - 2^1 = 125, 127 - 2^2 = 123, 127 - 2^3 = 119, 127 - 2^4 = 111, 127 - 2^5 = 95, and 127 - 2^6 = 63 are all composite. - _Michael B. Porter_, Aug 29 2016
%p N:= 10000: # to get all terms <= N
%p P:= select(isprime, {2,seq(i,i=3..N,2)}):
%p T:= {seq(2^i,i=0..ilog2(N))}:
%p R:= {seq(i,i=1..N,2)} minus {seq(seq(p+t,p=P),t=T)}:
%p sort(convert(R,list)); # _Robert Israel_, Sep 23 2016
%t Do[ i = 0; l = Ceiling[ N[ Log[ 2, n ] ] ]; While[ ! PrimeQ[ n - 2^i ] && i < l, i++ ]; If[ i == l, Print[ n ] ], {n, 1, 2000, 2} ]
%t Join[{1},Select[Range[5,1999,2],!MemberQ[PrimeQ[#-2^Range[Floor[ Log[ 2,#]]]], True]&]] (* _Harvey P. Dale_, Jul 22 2011 *)
%o (PARI) isA006285(n,i=1)={ bittest(n,0) && until( isprime(n-i) || n<i<<=1,); i>n } \\ _M. F. Hasler_, Jun 19 2008, updated Apr 12 2017
%o (Haskell)
%o a006285 n = a006285_list !! (n-1)
%o a006285_list = filter ((== 0) . a109925) [1, 3 ..]
%o -- _Reinhard Zumkeller_, May 27 2015
%o (Magma) lst:=[]; for n in [1..1973 by 2] do x:=-1; repeat x+:=1; a:=n-2^x; until a lt 1 or IsPrime(a); if a lt 1 then Append(~lst, n); end if; end for; lst; // _Arkadiusz Wesolowski_, Aug 29 2016
%o (Python)
%o from itertools import count, islice
%o from sympy import isprime
%o def A006285_gen(startvalue=1): # generator of terms
%o return filter(lambda n: not any(isprime(n-(1<<i)) for i in range(n.bit_length()-1,-1,-1)), count(max(startvalue+(startvalue&1^1),1),2))
%o A006285_list = list(islice(A006285_gen(),30)) # _Chai Wah Wu_, Nov 29 2023
%Y Cf. A133122, A098237, A065381, A156695, A109925, A118954, A232460, A276417, A254248.
%Y See also A058517, A350957, A350959, A350960.
%K nonn,nice,easy
%O 1,2
%A _N. J. A. Sloane_
%E More terms from Larry Reeves (larryr(AT)acm.org), Apr 13 2000
|
